Temperature factor anisotropic

Experimentally Uv may be obtained by fitting quantitatively the calculated X-ray beam intensities with the experimentally measured X-ray intensities using a general anisotropic temperature factor [42],... 

For proteins the X-ray structures usually are not determined at high enough resolution to use anisotropic temperature factors. Average values for B in protein structures range from as low as a few A2 for well-ordered structures to 30 A2 for structures involving flexible surface loops. Using equation 3.6, one can calculate the root mean square displacement fu2 for a well-ordered protein structure at approximately 0.25 A (for B = 5 A2) and for a not-so-well-ordered structure at... 

Least squares refinement of coordinates together with anisotropic temperature factors, in the final stages, gave an R factor of 0.11. 

For anisotropic vibration the temperature factor is more complex because f now depends on the direction of S. The anisotropic temperature factor is often... 

SHELXL (Sheldrick and Schneider, 1997) is often viewed as a refinement program for high-resolution data only. Although it undoubtedly offers features needed for that resolution regime (optimization of anisotropic temperature factors, occupancy refinement, full matrix least squares to obtain standard deviations from the inverse Hessian matrix, flexible definitions for NCS, easiness to describe partially... 

The anisotropic temperature factor will be the Fourier transform of P(u), given by... 

Positional and anisotropic parameters are given x 10. Numbers in parentheses are the estimated standard deviations in the units of the least significant figure given for the corresponding parameter. The anisotropic temperature factor is expl-tB h2 + B22-2 + e33 -2 + + gi3hA + 23 1 ... 

With small molecules, it is usually possible to obtain anisotropic temperature factors during refinement, giving a picture of the preferred directions of vibration for each atom. But a description of anisotropic vibration requires six parameters per atom, vastly increasing the computational task. In many cases, the total number of parameters sought, including three atomic coordinates, one occupancy, and six thermal parameters per atom, approaches or exceeds the number of measured reflections. As mentioned earlier, for refinement to succeed, observations (measured reflections and constraints such as bond lengths) must outnumber the desired parameters, so that least-squares solutions are adequately overdetermined. For this reason, anisotropic temperature factors for proteins have not usually been obtained. The increased resolution possible with synchrotron sources and cryocrystallography will make their determination more common. With this development, it will become possible to obtain better estimates of uncertainties in atom positions than those provided by the Luzzati method. 

Sorum [12] to a final R of 0.10. Thus, the reinvestigation of the structure of acetylcholine bromide, [C7H1602N] + Br, with X-ray diffraction intensities collected from two untwinned crystals showed that the crystals are monoclinic, and are characterized by a space group of P21/ra, with a = 10.966 (4), b = 13.729 (7), c = 7.159 (4) A, p = 108.18 (7)°, and Z = 4. The structure was refined by full-matrix least squares calculations using 1730 observed reflections, and anisotropic temperature factors for all non-hydrogen atoms. The final R was found to be 0.041. Atomic coordinates, thermal parameters, bond lengths, and angles were compared with those from a previous work on acetylcholine derivatives. 

A preliminary knowledge of the crystal structure is important prior to a detailed charge density analysis. Direct methods are commonly used to solve structures in the spherical atom approximation. The most popular code is the Shelx from Sheldrick [26] which provides excellent graphical tools for visualization. The refinement of the atom positional parameters and anisotropic temperature factors are carried out by applying the full-matrix least-squares method on a data corrected if found necessary, for absorption and diffuse scattering. Hydrogen atoms are either fixed at idealized positions or located using the difference Fourier technique. 

Results of Rietveld refinement of neutron powder data for Na-A zeolite at various temperatures above and below the cubic-rhombohedral transition. The space group used in each refinement is as shown. In the rhombohedral refinements, the positional parameters were constrained to satisfy Fm3c symmetry as discussed in text, f is the fractional occupancy and N the number of reflections included in this refinement. Also shown in columns I and II are the X-ray single crystal results of Pluth and Smith(t), but with their anisotropic temperature factors converted to the average isotropic values. Neutron... 

Although these effects are small, they affect not only the covalent X-H bond lengths, but also the H - A hydrogen-bond lengths. For very high-precision low-temperature neutron diffraction analyses [186, 187], they have to be taken into account. An adequate way to do this, without the complexity of a complete anhar-monic thermal motion analysis, is to use the experimentally determined anisotropic temperature factors. As shown by comparisons between experimental and theoretical X-H values from low temperature neutron analyses and theoretical ab-initio molecular orbital calculations, one can assume that the motion of the two... 

Final atomic coordinates and anisotropic temperature factors are listed in Table 16. Selected bond lengths and bond angles are listed in Table 17, along with the main intermolecular interatomic distances. 

In this case the temperature factor also becomes anisotropic, that is, reflections are weakened unequally in different direction. Historically, three different representations of anisotropic temperature factor have been used in the literature, namely,... 

Since the intensities of the standards were observed to diminish (finally to 85% of their original values) in a regular and nearly isotropic manner, the data were scaled linearly between each pair of standards. Associated with this decrease we also noted a decrease in the parameters b and y (which were in the end reduced by 0.02 A and 0.21 from their initial values). Broadening of the scans of file standards from 0.10 to 0.35 was also observed. The positions of the heavy atoms were determined from a three-dimensional Patterson synthesis. These positions were subjected to least-squares refinement as xenon atoms, after which it was possible to separate the antimony atoms by exploiting temperature factor differences. The positions were then further refined. A difference Fourier revealed positions for 12 of the 14 fluorine atoms. Least-.squares refinement of these positions was followed by another difference Fourier which revealed the positions of the final two fluorine atoms. Refinement of all these positions, with anisotropic temperature factors, resulted in a conventional/ factor of 0.06. Wei ting sch es were as previously described. ... 

Waser, J. The anisotropic temperature factor in triclinic coordinates. Ada Cryst. 8, 731 (1955). 

Hamilton, W. C. On the isotropic temperature factor equivalent to a given anisotropic temperature factor. Ada Cryst. 12, 609-610 (1959). 

It should be noted that more complicated forms of the temperature factor term can be employed when the crystal structure analysis is particularly precise and the resolution high. These expressions take into account the possible anisotropy of the thermal motion or statistical disorder. In the most sophisticated cases, six parameters are used to define the three-dimensional ellipsoids of thermal motion, which serve to describe anisotropic temperature factors. These should not be a source of concern to the reader at this time. 

In the very highest-resolution structures, we can allow deviations from this approximation, using something called anisotropic temperature factors. This describes the motions of the atoms by tensors, which are outside the scope of this chapter. For a description, see Schneider22. 

T. R. Schneider, What can we Learn from Anisotropic Temperature Factors In Proceedings of the CCP4 Study Weekend, 1996 pp 133-144. 

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